Duke Mathematical Journal

Homogeneous asymptotic limits of Haar measures of semisimple linear groups and their lattices

François Maucourant

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We show that Haar measures of connected semisimple groups, embedded via a representation into a matrix space, have a homogeneous asymptotic limit when viewed from far away and appropriately rescaled. This is still true if the Haar measure of the semisimple group is replaced by the Haar measure of an irreducible lattice of the group, and the asymptotic measure is the same. In the case of an almost simple group of rank greater than 2, a remainder term is also obtained. This extends and makes precise anterior results of Duke, Rudnick, and Sarnak [DRS] and Eskin and McMullen [EM] in the case of a group variety

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Duke Math. J., Volume 136, Number 2 (2007), 357-399.

First available in Project Euclid: 21 December 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E45: Representations of Lie and linear algebraic groups over real fields: analytic methods {For the purely algebraic theory, see 20G05} 11P21: Lattice points in specified regions
Secondary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]


Maucourant, François. Homogeneous asymptotic limits of Haar measures of semisimple linear groups and their lattices. Duke Math. J. 136 (2007), no. 2, 357--399. doi:10.1215/S0012-7094-07-13626-3. https://projecteuclid.org/euclid.dmj/1166711374

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